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5.2.7 Parametrized Covariant Transport

Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. To every morphism $f: C \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{C}}$, Definition 5.2.2.1 associates a covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$, which is uniquely determined up to isomorphism (see Proposition 5.2.2.4). Our goal in this section is to show that the functor $f_{!}$ can be chosen to depend functorially on the morphism $f$: that is, the construction $f \mapsto f_{!}$ can be promoted to a functor from the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$. We begin by introducing a more elaborate version of Definition 5.2.2.1.

Definition 5.2.7.1 (Parametrized Covariant Transport). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibration of simplicial sets, and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. We will say that a morphism $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by parametrized covariant transport if there exists a morphism of simplicial sets $\widetilde{F}: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:

$(1)$

The diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \widetilde{F} } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \ar [r] & \operatorname{\mathcal{C}}} \]

commutes (where the lower horizontal map is induced by the inclusion $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \hookrightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).

$(2)$

The restriction $\widetilde{F}|_{ \{ 0\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}}$ is given by projection onto $\operatorname{\mathcal{E}}_{C}$, and the restriction $\widetilde{F}|_{ \{ 1\} \times C^{\ast }(\operatorname{\mathcal{E}}) }$ is equal to $F$.

$(3)$

For every edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ and every object $X \in \operatorname{\mathcal{E}}_{C}$, the composite map

\[ \Delta ^1 \times \{ f\} \times \{ X\} \hookrightarrow \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_ C \xrightarrow { \widetilde{F} } \operatorname{\mathcal{E}} \]

is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.

If these conditions are satisfied, we say that the morphism $\widetilde{F}$ witnesses $F$ as given by parametrized covariant transport.

Remark 5.2.7.2. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$, and let $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be given by parametrized covariant transport. Then, for every edge $f: C \rightarrow D$, the composite map

\[ \{ f\} \times \operatorname{\mathcal{E}}_{C} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \xrightarrow {F} \operatorname{\mathcal{E}}_{D} \]

is given by covariant transport along $f$, in the sense of Definition 5.2.2.1. In other words, we can identify $F$ with a diagram $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$, which carries each edge $f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the covariant transport functor $f_{!}$ of Notation 5.2.2.5.

Example 5.2.7.3. Let $\operatorname{Set}_{\ast }$ denote the category of pointed sets (Example 4.2.3.3), and let $V: \operatorname{Set}_{\ast } \rightarrow \operatorname{Set}$ denote the forgetful functor $(X,x) \mapsto X$. Then the induced map $\operatorname{N}_{\bullet }(V): \operatorname{N}_{\bullet }( \operatorname{Set}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$ is a cocartesian fibration (in fact, it is a left covering map), whose fiber over an object $X \in \operatorname{N}_{\bullet }(\operatorname{Set})$ can be identified with the set $X$. For every pair of sets $X$ and $Y$, the evaluation map

\[ \operatorname{ev}: \operatorname{Hom}_{\operatorname{Set}}(X,Y) \times X \rightarrow Y \quad \quad (f,x) \mapsto f(x) \]

is given by parametrized covariant transport (in the sense of Definition 5.2.7.1).

Proposition 5.2.7.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. Then:

  • There exists a morphism $F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ which is given by parametrized covariant transport.

  • An arbitrary diagram $F': \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by parametrized covariant transport if and only if it is isomorphic to $F$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$).

Proof. Apply Lemma 5.2.2.9 to the simplicial set $K = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}$.. $\square$

Remark 5.2.7.5 (Functoriality). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^-{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{\overline{G}} & \operatorname{\mathcal{C}}', } \]

where $U$ and $U'$ are cocartesian fibrations. Let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$ having images $C' = \overline{G}(C)$ and $D' = \overline{G}(D)$, respectively, so that $G$ induces functors $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$ and $G_{D}: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}'_{D'}$. Let $\varphi : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(C',D')$ be the morphism induced by $\overline{G}$, and let

\[ F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D} \quad \quad F': \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(C',D') \times \operatorname{\mathcal{E}}'_{C'} \rightarrow \operatorname{\mathcal{E}}'_{D'} \]

be given by parametrized covariant transport with respect to $U$ and $U'$. Suppose that the morphism $G$ carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to $U'$-cocartesian edges of $\operatorname{\mathcal{E}}'$. Then the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{F} \ar [d]^{ \varphi \times G_{C} } & \operatorname{\mathcal{E}}_{D} \ar [d]^{G_ D} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}'}(C',D') \times \operatorname{\mathcal{E}}'_{C'} \ar [r]^-{F'} & \operatorname{\mathcal{E}}'_{D'} } \]

commutes up to isomorphism: that is, $G_{D} \circ F$ and $F' \circ (\varphi \times G_{C})$ are isomorphic as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}'_{D'} )$. This follows by applying the uniqueness assertion of Lemma 5.2.2.9 to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r] \ar [d] & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{C}}'. } \]

Variant 5.2.7.6 (Parametrized Contravariant Transport). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $C$ and $D$ be vertices of $\operatorname{\mathcal{C}}$. Applying Proposition 5.2.7.4 to the opposite cocartesian fibration $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a diagram $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$, carrying each edge $f: C \rightarrow D$ to a functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ given by contravariant transport along $f$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with the fiber over $Y$ of the left fibration $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ of Proposition 4.6.4.10, or with the fiber over $X$ of the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $. In either case, parametrized transport recovers the composition law of $\operatorname{\mathcal{C}}$:

Proposition 5.2.7.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $C$, $D$, and $E$. Then the composition law

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \]

of Construction 4.6.7.9 is given by parametrized covariant transport for the left fibration $U: \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 5.2.7.1), and also by parametrized contravariant transport for the right fibration $V: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ E\} \rightarrow \operatorname{\mathcal{C}}$.

Proof. We will prove the first assertion; the second follows by a similar argument. Let $S: \Delta ^1 \times \Delta ^1 \rightarrow \Delta ^2$ be the morphism given on vertices by the formula $T(i,j) = i(j+1)$, and let $T$ be a section of the trivial Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ (see Corollary 4.6.7.5). Then the composite map

\[ \Delta ^1 \times \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \xrightarrow {S \times T} \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{\mathcal{C}} \]

carries $\{ 0\} \times \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ to the vertex $C$, and can therefore be identified with a functor

\[ \widetilde{F}: \Delta ^1 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}. \]

Unwinding the definitions, we see that $\widetilde{F}$ exhibits the composition law as given by covariant transport along the identity map $\operatorname{id}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E)$ for the left fibration $U$. $\square$

Proposition 5.2.3.1 has a counterpart for parametrized covariant transport:

Proposition 5.2.7.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Let $C$, $D$, and $E$ be objects of $\operatorname{\mathcal{C}}$, and let

\[ F: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D} \quad \quad G: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{E} \]
\[ H: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{E} \]

be given by parametrized covariant transport. Then the diagram

5.26
\begin{equation} \begin{gathered}\label{equation:transitivity-parametrized} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ \operatorname{id}\times F } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \ar [d]^{ G } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ H } & \operatorname{\mathcal{E}}_{E} } \end{gathered} \end{equation}

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$; here the left vertical map is given by the composition law of Construction 4.6.7.9.

Proof. Let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E)$ be the Kan complex defined in Notation 4.6.7.1, let $H'$ denote the composite map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \xrightarrow {H} \operatorname{\mathcal{E}}_{E}, \]

and let $H''$ denote the composition

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \\ & \xrightarrow {F} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \\ & \xrightarrow {G} & \operatorname{\mathcal{E}}_{E}. \end{eqnarray*}

We will show that $H'$ and $H''$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{E} )$. The homotopy commutativity of the diagram (5.26) will then follow by precomposing with any section of the trivial Kan fibration $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$.

Choose morphisms

\[ \widetilde{F}: \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]

\[ \widetilde{G}: \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}} \]

\[ \widetilde{H}: \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]

which witness $F$, $G$, and $H$ as given by parametrized covariant transport, respectively. Composing with the projection maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \leftarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E), \]

we obtain morphisms

\[ \widetilde{F}': \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}} \]

\[ \widetilde{H}': \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}. \]

Let $\widetilde{G}'$ denote the composite map

\begin{eqnarray*} \operatorname{N}_{\bullet }( \{ 1 < 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \hspace{-.5em} & \rightarrow & \operatorname{N}_{\bullet }( \{ 1 < 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \\ & \xrightarrow { F } & \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E) \times \operatorname{\mathcal{E}}_{D} \\ & \xrightarrow { \widetilde{G} } & \operatorname{\mathcal{E}}. \end{eqnarray*}

Since $U$ is an inner fibration, the lifting problem

\[ \xymatrix@C =100pt@R=50pt{ \Lambda ^{2}_{1} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \ar [r]^-{ ( \widetilde{G}', \bullet , \widetilde{F}' ) } \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \ar [r] \ar@ {-->}[ur]^{\Phi } & \operatorname{\mathcal{C}}} \]

admits a solution $\Phi : \Delta ^2 \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}$. Let $\widetilde{H}''$ denote the restriction of $\Phi $ to the product $\operatorname{N}_{\bullet }( \{ 0, 2\} ) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}$. Using Proposition 5.1.4.12, we see that $\widetilde{H}''$ is a $U$-cocartesian lift of $U \circ \widetilde{H}'' = U \circ \widetilde{H}'$, in the sense of Definition 5.2.2.6. Applying the uniqueness assertion of Lemma 5.2.2.9, we conclude that the restrictions $H' = \widetilde{H}'|_{ \{ 2\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} }$ and $H'' = \widetilde{H}''|_{ \{ 2\} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C} }$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D,E) \times \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{E})$, as desired. $\square$

Using Proposition 5.2.7.8, we obtain the following refinement of Construction 5.2.3.2:

Construction 5.2.7.9 (Enriched Homotopy Transport: Covariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (Construction 4.6.7.13). It follows from Proposition 5.2.7.8 (and Example 5.2.2.2) that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:

  • For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map

    \[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )^{\simeq } \]

    in $\mathrm{h} \mathit{\operatorname{Kan}}$ corresponds to the parametrized covariant transport functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \times \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ of supplied by Proposition 5.2.7.4 (which is well-defined up to isomorphis).

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cocartesian fibration $U$. Note that the underlying functor of ordinary categories $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ coincides with homotopy transport representation of Construction 5.2.3.2.

Remark 5.2.7.10 (Functoriality). Suppose we are given a commutative diagram of $\infty $-categories

5.27
\begin{equation} \begin{gathered}\label{equation:enriched-homotopy-transport-functorial} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r]^-{G} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{G} } & \operatorname{\mathcal{C}}', } \end{gathered} \end{equation}

where $U$ and $U'$ are cocartesian fibrations and the functor $G$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ to $U'$-cocartesian morphisms of $\operatorname{\mathcal{E}}'$. For each object $C \in \operatorname{\mathcal{C}}$ having image $C' = \overline{G}(C)$, $G$ restricts to a functor $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C'}$. It follows from Remark 5.2.7.5 that the construction $C \mapsto G_{C}$ determines a natural transformation of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors $\alpha : \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \operatorname{hTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}'} \circ \mathrm{h} \mathit{\overline{G}}$ from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to $\mathrm{h} \mathit{\operatorname{QCat}}$. Moreover, if (5.27) is a pullback square, then $\alpha $ is an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors.

Variant 5.2.7.11 (Enriched Homotopy Transport: Left Fibrations). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. Applying Construction 5.2.7.9, we obtain an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}, \]

given on objects by the formula $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}( C ) = \{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Variant 5.2.7.12 (Enriched Homotopy Transport: Contravariant Case). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of $\infty $-categories. Applying Construction 5.2.7.9 to the opposite functor $U^{\operatorname{op}}$, we deduce that there is a unique $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ with the following properties:

  • For each object $C$ of the $\infty $-category $\operatorname{\mathcal{C}}$, $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ (regarded as an object of $\mathrm{h} \mathit{\operatorname{QCat}}$).

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, the induced map

    \[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } \]

    is given by the parametrized contravariant transport functor $\operatorname{\mathcal{E}}_{D} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,C) \rightarrow \operatorname{\mathcal{E}}_{C}$ of Variant 5.2.7.6.

We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the enriched homotopy transport representation of the cartesian fibration $U$. If $U$ is a right fibration, then $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ takes values in the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$.

Example 5.2.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category, which we regard as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes. Applying Proposition 5.2.7.7, we obtain the following:

  • For every object $C \in \operatorname{\mathcal{C}}$, the corepresentable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

    \[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]

    is the enriched homotopy transport representation for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

  • For every object $D \in \operatorname{\mathcal{C}}$, the representable $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor

    \[ \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}} \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]

    is the enriched homotopy transport representation for the right fibration $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ D\} \rightarrow \operatorname{\mathcal{C}}$.